# The Dirac Delta Function Potential

### From FSUPhysicsWiki

A delta function potential, , is a special case of the symmetric finite square well; it is the limit in which the width of the well goes to zero and the depth of the well goes to infinity, while the product of the height and depth remains constant. For such a potential, the wave function is still continuous across the potential, ie. . However, the first derivative of the wave function is not.

For a particle subject to an attractive delta function potential, the Schrödinger equation is

For the potential term vanishes, and all that is left is

One or more bound states may exist, for which , and vanishes at . The bound state solutions are given by

where

The first boundary condition, the continuity of at , yields .

The second boundary condition, the discontinuity of at , can be obtained by integrating the Schrödinger equation from to and then letting

Integrating the whole equation across the potential gives

In the limit , we have

which yields the relation, .

Since we defined , we have . Then, the energy is

Finally, we normalize :

so,

Evidently, the delta function well, regardless of its "strength" , has one bound state:

For a delta potential of the form , we may apply a similar procedure to the one employed above to show that the discontinuity in the derivative of the wave function is

## Problem

(Double delta function potential)

Consider a double delta function potential, Prove that a ground state with even parity always exists. By means of a sketch, show that a first excited state also exists for a sufficiently large potential strength and sufficiently large well separation